Free Resolutions

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• 批准号: 0900931
• 负责人: Irena Peeva
• 金额: $ 10.75万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900931&HistoricalAwards=false
• 关键词: Free; Resolutions

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project is in the field of Commutative Algebra. The proposed research deals with the structure of graded free resolutions, their numerical invariants, and applications. The general research goal is to study graded free resolutions using a variety of methods from Commutative Algebra, Computational Algebra, and Topological Combinatorics. The project focuses on: Borel ideals and applications of mapping cones over Clements-Lindstrom rings, resolutions of monomial edge ideals, and infinite cellular resolutions. The proposed research involves interdisciplinary approaches and connects Commutative Algebra with the fields of Combinatorics, Computational Algebra, and Topology.The idea to associate a free resolution to a finitely generated module was introduced by Hilbert in two famous papers in 1890 and 1893. He proved that over a polynomial ring (over a field) every finitely generated module has a finite free resolution. If the ring and the module are graded then there exists a minimal free resolution; it is unique up to an isomorphism and is contained in any free resolution. The minimal free resolution is graded, and its properties are closely related to the invariants of the module. From another point of view: in essence constructing a free resolution consists of repeatedly solving systems of linear equations. Recent computational methods have made it possible to compute graded free resolutions by computer. For many years, free resolutions have been both central objects and fruitful tools in Commutative Algebra; they have many applications in other mathematical fields.

该奖项由 2009 年美国复苏和再投资法案（公法 111-5）资助。该项目属于交换代数领域。拟议的研究涉及分级自由分辨率的结构、它们的数值不变量和应用。一般研究目标是使用交换代数、计算代数和拓扑组合学的各种方法来研究分级自由分辨率。该项目重点关注：Borel 理想和在 Clements-Lindstrom 环上映射锥体的应用、单项式边缘理想的分辨率以及无限细胞分辨率。拟议的研究涉及跨学科方法，并将交换代数与组合学、计算代数和拓扑学领域联系起来。将自由解析与有限生成模相关联的想法是希尔伯特在 1890 年和 1893 年的两篇著名论文中提出的。他证明了在多项式环上（在域上），每个有限生成的模块都具有有限的自由分辨率。如果环和模块是分级的，则存在最小自由分辨率；它在同构上是唯一的，并且包含在任何自由分辨率中。最小自由分辨率是分级的，其性质与模块的不变量密切相关。从另一个角度来看：构建自由分辨率本质上是由重复求解线性方程组组成。最近的计算方法使得通过计算机计算分级自由分辨率成为可能。多年来，自由解析一直是交换代数的中心对象和富有成效的工具。它们在其他数学领域有许多应用。